A natural resource such as oil or gas residing in a subterranean formation can be recovered by drilling a well into the formation. The subterranean formation is usually isolated from other formations using a technique known as cementing. In particular, a wellbore is typically drilled down to the subterranean formation while circulating a drilling fluid through the wellbore. After the drilling is terminated, a string of pipe (e.g. casing string) is run in the wellbore. Primary cementing is then usually performed whereby a cement slurry is pumped down through the casing string and into the annulus between the casing string and the wall of the wellbore or another casing string to allow the cement slurry to set into an impermeable cement column and thereby fill a portion of the annulus. Sealing the annulus typically occurs near the end of cementing operations after well completion fluids, such as spacer fluids and cements, are trapped in place to isolate these fluids within the annulus from areas outside the annulus. The annulus is conventionally sealed by closing a valve, energizing a seal, and the like.
In addition to the uniform exterior loads placed on the casing, non-uniform exterior loads are also placed on the casing as a result of pressure loading from salt and overburden loading in horizontal wells, for example. Overburden and lateral formation pressures cause salt to flow, closing the wellbore outside the casing. Because the lateral formation pressures may not be equal, or because the wellbore is not perfectly circular, the loading from the salt will not, in general, be uniform. In a horizontal well, overburden loads may cause the wellbore to deform vertically, contacting the casing and producing a pressure load.
Conventional well known finite element analysis techniques have been used to test the casing design for structural integrity when the casing is under non-uniform exterior loads, however, such techniques are complex, time consuming, and may not be appropriate for a general casing design program. Other, less popular, techniques use the equations of equilibrium for a shell to test the casing design for structural integrity when the casing is under non-uniform exterior loads. The loading is discontinuous and equal to a positive load over an angle θ<θ1, where θ1 has been determined from experimental data to be π/20 or about 9°. The solution is obtained using the principle of virtual work and the conservative assumption that the radius of the shell is equal to outer radius rather than the average radius. Bending, hoop, and shear stresses may be obtained from the solution, since bending moment, circumferential force, and shear force are determined from the equilibrium equations. Because radial loads are not considered in this technique, the solution is merely an approximation. Moreover, because this technique assumes a thin wall radius, it is generally unacceptable for testing the structural integrity of most casing designs.